Suppose
g(x)={1/x+1 if x<1
{2x-1 if x≥1
The best description concerning the continuity of g(x) is that the function:
is continuous.
has a jump discontinuity.
has an infinite discontinuity.
has a removable discontinuity.
has both jump and infinite discontinuity.
infinite at x = -1
also, a jump at x=1
lim x->1- = 1/2
but g(1) = 1
Would it be infinite?
To determine the continuity of the function g(x), we need to examine the behavior of the function at the point where x = 1, since this is where the definition of g(x) changes.
Looking at the two pieces of the function separately:
For x < 1, g(x) = 1/(x + 1). This is a rational function and is continuous for all values of x except where the denominator is zero. However, x + 1 is never zero, so this part of the function is continuous for all x < 1.
For x ≥ 1, g(x) = 2x - 1. This is a linear function and is continuous for all values of x.
Since both pieces of the function are continuous individually, we need to determine whether g(x) is continuous at x = 1 by checking if the limit from the left is equal to the limit from the right as x approaches 1.
The left limit of g(x) as x approaches 1 is:
lim(x→1-) g(x) = lim(x→1-) (1/(x + 1)) = 1/2
The right limit of g(x) as x approaches 1 is:
lim(x→1+) g(x) = lim(x→1+) (2x - 1) = 2 - 1 = 1
Since the left limit and the right limit are not equal, the function g(x) has a jump discontinuity at x = 1.
Therefore, the best description concerning the continuity of g(x) is that the function has a jump discontinuity.